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20
votes
4answers
2k views

Eigenvalues of permutations of a real matrix: can they all be real?

For a matrix $M\in GL(n,\mathbb R)$, consider the $n!$ matrices obtained by permutations of the rows (say) of $M$ and define the total spectrum $TS(M)$ as the union of all their spectra (counting ...
9
votes
1answer
1k views

Frobenius-Perron eigenvalue and eigenvector of sum of two matrices

Suppose that I have two positive matrices, $A$, and $B$, and I know their Frobenius-Perron eigenvalues ($\lambda_A$, $\lambda_B$) and eigenvectors ($v_A$, $v_B$). I'm interested in what I can say ...
8
votes
0answers
417 views

Eigenvalues of permutations of a real matrix: how complex can they be?

This is sort of complementary to this thread. I’ll repeat the definitions here: For a matrix $M\in GL(n,\mathbb R)$, consider the $n!$ matrices obtained by permutations of the rows (say) of $M$ and ...
7
votes
7answers
639 views

Semicircle law universality elsewhere

Wigner's semicircle distribution is: $$f(x)=\frac{1}{2 \pi}\sqrt{4-x^2}, \ \ -2\leq x\leq 2.$$ Under reasonable conditions, the rescaled eigenvalue density of random symmetric matrices $M_n$ follows ...
7
votes
1answer
278 views

Can I find the gap between the two least eigenvalues of this special matrix A(t)?‎

I am interested in finding the gap between the two least eigenvalues of $A(t)$, a Hermitian $N\times N$ sparse ‎matrix whose diagonal elements are $a_it+b_i\,(1\leq i\leq N)$, and all off-diagonal ...
7
votes
0answers
153 views

Toda Flow Embeddings

What are strategies for generating the following types of pictures: Here's what's going on here. Take a toda flow in 3 variables. The equations of motion are: ...
6
votes
2answers
342 views

Does small Perron-Frobenius eigenvalue imply small entries for integral matrices?

Suppose that $M$ is an $n \times n$ matrix where each entry is a positive integer. Then $M$ is Perron-Frobenius and so has unique largest real eigenvalue $\lambda_{\textrm{PF}}$. Does an upper ...
6
votes
5answers
534 views

The maximal eigenvalue of a symmetric Toeplitz matrix

Let $0\le x\le 1$ be a real number. Denote by $A_n(x)=(a_{ij})$ the $n$ by $n$ matrix such that $a_{ij}=x^{|i-j|}$ and let $\lambda_n(x)$ be the maximal eigenvalue of $A_n(x)$. Is there any ...
6
votes
1answer
232 views

Why are some solutions of these diophantine equations off the usual patterns?

This is inspired by a recent question about complete multipartite integral graphs. I am wondering if more can be said about tripartite integral graphs with block sizes $a<b<c$. It is easy to see ...
6
votes
1answer
931 views

Proving that a specific kernel is positive definite

Most theoretical papers concerning kernels assume that they are given a positive definite kernel. In this question, we want to show that a specific kernel is positive definite. We are interested in ...
6
votes
1answer
135 views

Extension of Wigner's semicircle law?

It is well-known that the semicircle law holds for a wide class of matrices with independent and identically distributed (mean zero) entries. My question is: is there any study about the more general ...
6
votes
1answer
176 views

Eigenvalue inequality for regular graphs

I recently proved an inequality relating some of the eigenvalues of a regular graph with each other, and I was wondering if it is already known. I was unable to find it online, and a quick skim ...
5
votes
1answer
142 views

Eigenvalues of $X$ in the metric of $Y$

What does this statement describe? $X$ and $Y$ are matrices. The eigenvalues of $X$ in the metric of $Y$. I've not seen this language used before in this fashion and I don't really know what ...
5
votes
1answer
338 views

Anti-bidiagonal matrix with main anti-diagonal {1,2,3,…} and first sub-anti-diagonal {-1,-2,-3,…} has eigenvalues lambda={1,-2,3,-4,…}

Consider the anti-bidiagonal matrix $B_6\in\mathbb{R}^{6\times 6}$, defined along its anti-diagonals as follows $$ B_6=\begin{bmatrix} & & & & & 6\\ & & & ...
5
votes
1answer
136 views

Sum of the absolute eigenvalues of A>=B

Kindly help me to prove/disprove the following statement. Let $A$ be a symmetric matrix of order $n \times n$ with all the diagonal entry equal to $0$, and other non-diagonal entry equal to $k$ ...
5
votes
2answers
592 views

Linearly constrained eigenvalue problem

Suppose I'd like to: \begin{align} \mathop{\text{min}}_\mathbf{x} && \mathbf{x}^T\mathbf{A}\mathbf{x} \\ \text{subject to:} && \mathbf{x}^T \mathbf{M} \mathbf{x} = 1\\ && ...
5
votes
2answers
254 views

No Exceptional Eigenvalues of Weight 1/2 Maass Forms on $\Gamma_0(4)$?

Some colleagues and I were wondering if there is a citation out there which shows there are no exceptional eigenvalues, $\lambda$, of classical weight 1/2 Maass forms on $\Gamma_0(4)$, which is to say ...
5
votes
2answers
215 views

Probabilistic Interpretation of First Dirichlet Eigenvalue?

The first Dirichlet eigenvalue of a compact domain $\Omega\subset\mathbb{R}^n$ with smooth boundary is the smallest positive number for which there exists a non-trivial solution to $$ -\Delta\psi = ...
5
votes
2answers
1k views

Eigenvalues of a Symmetric Positive Semi-Definite (PSD) matrix after rank one update

I have a Symmetric Positive Semi-Definite matrix $A$ which i know its eigenvalue and eigenvectors. let $v$ and $u$ be a random column vector. i want to know if it is possible to have eigenvalues of ...
5
votes
1answer
230 views

Separating the spectrum of a Hermitian matrix

Given Hermitian matrix $A$, I would like to perturbate it so that its eigenvalues become well-separated. Specifically, let $A$ be some Hermitian matrix, and let $G$ be a Gaussian matrix, with each ...
5
votes
0answers
184 views

Eigenvalues of a certain product of matrices with special structure

Let $d$ and $c$ be positive integers and $q = dc$. Let $G$ be a $q$-by-$q$ positive semi-definite real matrix with eigenvalues all $\le 1$, and define the $q$-by-$2q$ matrix $A = ...
5
votes
0answers
118 views

Possible values of eigenvalues of Hadamard product of Hermitian matrices

One of the most important (and very well-known) result in the study of the spectrum of Hermitian matrices is Horn's conjecture (or theorem?), which provides a complete answer to the following problem: ...
5
votes
0answers
92 views

Eigenvalues of Random Regular Bipartite Graphs

I am looking for a way of getting a good estimate of the eigenvalues of random bipartite d-regular graphs. The literature has very precise values the proofs of which are very involved and since I am ...
5
votes
0answers
211 views

Existence of a matrix product from its eigenvalues

Let A and B be two positive definite, real, symmetric matrices. The eigenvalues of A, B and AB, denoted by $\lambda(X)$, obey the relation (from Bhatia): $$ \lambda^\downarrow(A) \cdot ...
4
votes
3answers
254 views

How networks with high largest eigenvalues are more robust?

In the literature, it is sometimes indicated that network with high value of largest eigenvalue (either adjacency matrix or its Laplacian counterpart) are more robust against link/node removals. ...
4
votes
2answers
127 views

Relation between eigenvalues of $A$ and $A^TA$?

For an $n\times n$ diagonizable matrix $A$, is there a relation between the eigenvalues of $A$ and the eigenvalues of $A^TA$? I ask this because I am looking into the relation between $A$ and $A+cI$, ...
4
votes
2answers
232 views

Is there an easy way to tell if all eigenvalues of a unitary or self-adjoint matrix only have eigenvalues of multiplicity two?

I am interested in a class of $2n\times 2n$ unitary matrices with complex entries (if you prefer, we can replace "unitary" with "self-adjoint"). I know that all the eigenvalues of matrices in this ...
4
votes
1answer
313 views

the eigenvalues of a generalized circulant matrix

A $2k\times 2k$ circulant matrix $\ C$ takes the form \begin{align} C= \begin{bmatrix} c_0 & c_{2k-1} & \dots & c_{2} & c_{1} \\ c_{1} & c_0 & c_{2k-1} & & c_{2} ...
4
votes
3answers
232 views

Is this function well studied?

Let $A_1,\dots,A_L$ be $N\times N$ hermitian matrices. Define the simplex \begin{align} \mathcal{S}=\left\{[x_1,\dots,x_L]\mid x_i\geq 0,~\sum_{i=1}^{L}x_i=1 \right\} \end{align} and consider the ...
4
votes
2answers
149 views

Non-asympototic version of Gelfand's formula

Let $A$ be a $n\times n$ matrix. Let $\|A\|$ be the spectral norm of $A$, and $\rho(A)$ be the spectral radius. I am wondering whether the following statement is true. There exists universal ...
4
votes
1answer
101 views

An inequality for eigenvalues of the Dirichlet problem

Is either of these inequalities true? $$\lambda(tA + (1-t)B)\geq t\lambda(A) + (1-t)\lambda(B)$$ or $$\lambda(tA + (1-t)B)\leq t\lambda(A) + (1-t)\lambda(B),$$ where $0\leq t \leq 1$, $A,B$ are ...
4
votes
1answer
243 views

Lower bounds on matrix eigenvalues

Let $A$ be a real $n\times n$ matrix and let $\mu_1, \dots, \mu_n$ the (generalized, complex) eigenvalues of $A$. Assume that $$ 0 < \alpha < \mathrm{Re}(\mu_1) < \dots < ...
4
votes
1answer
956 views

Connection between eigenvalues of matrix and its Laplacian.

Hello! There are two definitions of graph spectrum: 1) Eigenvalues of adjacency matrix $A$. 2) Eigenvalues of Laplacian of adjacency matrix ($L$). Different sources offer different properties based ...
4
votes
1answer
162 views

Epidemic threshold

Need some help / ideas to proceed. Stuck for a while on this. In the literature of epidemic theory, it is found that the epidemic threshold is $1/\lambda_{max}(A)$ where $\lambda_{max}(A)$ is the ...
4
votes
2answers
1k views

rank-one perturbation of a matrix corresponding to a specific spectrum

Let $A$ be a real symmetric matrix whose spectrum is $\lambda_1,\lambda_2,\ldots,\lambda_n$. Let $A'$ be the matrix obtained by adding a perturbation to $A$. The requirement is that only the second ...
4
votes
1answer
137 views

Is the sum of spectral projections a projection?

Let $T$ be a closed operator on a Hilbert space with discrete spectrum. Then for $\{\lambda_1,...\lambda_n\}\in\sigma(T)$ one can define the spectral projections ...
4
votes
1answer
230 views

Question about a (relatively simple looking) differential operator and its eigenvalues

A colleague and I are interested in a specific differential operator on the reals. The differential operator L is of the form $L=-(1+x^{2})\frac{d^{2}}{dx^{2}}+c_{1}x\frac{d}{dx}+c_{2}x^{2}$ for ...
4
votes
1answer
231 views

A complex sequence with positive values

Let $\lambda_1,\dots,\lambda_d$ be complex numbers that constitute the spectrum of a nonnegative integer matrix, and $P_1,\dots, P_d$ be complex polynoms, such that the sequence $$u_n=\Sigma_{i=1}^d ...
4
votes
0answers
90 views

Classes for which the Spectrum determines a Convex Shape

Given a planar domain $\Omega \subset \Bbb{R}^2$ bounded and open we can associate to it the spectrum of the Laplace operator with Dirichlet boundary condition. It is known that there are planar ...
3
votes
5answers
6k views

Eigenvalues of Symmetric Tridiagonal Matrices

Suppose I have the symmetric tridiagonal matrix: $ \begin{pmatrix} a & b_{1} & 0 & ... & 0 \\\ b_{1} & a & b_{2} & & ... \\\ 0 & b_{2} & a & ... & 0 ...
3
votes
3answers
136 views

Can a block matrix with at least 3 zero blocks of different size on the diagonal and 1's everywhere else have only integer eigenvalues?

Let $M=\begin{pmatrix} \begin{array}{cccccccc} 0 & 0 & 1 & 1 & 1 & 1 & 1 &1\\ 0 & 0 & 1 & 1 & 1 & 1 & 1 &1\\ 1 & 1 & 0 & 0 & ...
3
votes
2answers
373 views

Prove log of eigenvalues are dense in R?

Suppose you have the set of all possible $n$ x $n$ square adjacency matrices where $n$={1,2,3,4...}. For each matrix, compute the logarithm of the largest eigenvalue. Is it true that the set of ...
3
votes
2answers
2k views

Upper bounds on eigenvalues of PSD matrix?

Suppose A is a symmetric positive semidefinite matrix. Is there a way to upper bound the largest eigenvalue using properties of its row sums or column sums? For instance, the Perron–Frobenius ...
3
votes
1answer
626 views

Why are 1 and -1 eigenvalues of this matrix?

This is a subject I've been working on for a very long time now, but still did not manage to fully understand the interesting properties of this matrix $\mathbf{A}$. First, let's define two matrices: ...
3
votes
1answer
349 views

Matrix norms / eigenvalues / singular values / another thing

OK, here is what is probably a stupid question. Let $M$ be a non-symmetric real matrix: for example, the shear matrix $\left( \begin{array}{cc} 1 & 1 \\\ 0 & 1 \end{array} \right)$. There ...
3
votes
2answers
614 views

What is known about the spectrum of a Cauchy matrix?

Math people: A Cauchy matrix is an $m$-by-$n$ matrix $A$ whose elements have the form $a_{i,j} = \frac{1}{x_i-y_j}$, with $x_i \neq y_j$ for all $(i, j)$, and the $x_i$'s and $y_i$'s belong to a ...
3
votes
1answer
111 views

Eigenfunctions to 2nd-order Differential Operators: Relation between Frobenius Series Solution and Eigenfunction Normalised to the Delta Function

Consider the 2nd-order linear ODE $x f^{''}(x) + x (\beta - 2 \alpha x) \kappa / \sigma f^{'}(x) - 1 / \sigma \left[ 2 \alpha \kappa - \lambda^2 (\beta - 2 \alpha x)^2 \right] f(x) = 0$, where ...
3
votes
1answer
210 views

When is this matrix singular?

Consider matrix $A$ with $(j,k)$′th entry $A_{j,k}=\sin(\omega_j t_k+\phi_j),\,\forall j,k\in\{1,2,...,n\}$, where $\omega_j,t_k,\phi_j\in \mathbf R$. 1) For $t_k=k$, what is the condition on ...
3
votes
1answer
139 views

Determinant of a Certain Positive-Definite Block Matrix

Is there a lower bound for the determinant or minimum eigenvalue of the following $d$ by $d$ matrix in terms of $d$? $$\Gamma=\left( {\begin{array}{cc} I & B \\ B^{*} & I \\ \end{array} ...
3
votes
1answer
277 views

Eigenvectors as continuous functions of matrix - diagonal perturbations

The general question has been treated here, and the response was negative. My question is about more particular perturbations. The counterexamples given in the previous question have variations not ...